Relativistic Constant Acceleration

Source

Lecture: 15 — Relativistic dynamics (four-acceleration, four-force, equations of motion).
Simulation: Relativistic constant acceleration — guided-inquiry mode.
Concept: Under a constant applied force a relativistic particle's velocity asymptotically approaches c rather than growing without bound, while its energy increases without limit.
Serves: Formulate Newton's 2nd law in Lorentz-invariant form (F = dp/dt, p = γmv); relate relativistic force to changes in four-momentum; see how the geometric speed limit c emerges dynamically from the equation of motion.
Inquiry: Before viewing the graph: what do you expect the v-vs-t graph to look like after a long time?
Misconception: "A constant force means constant acceleration, so velocity keeps increasing forever — there's no speed limit in F = ma." Encoded as prediction choice (A) and as the diverging dashed Newtonian curve.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Every element serves a single idea: the speed limit emerging from F = dp/dt.
A2	Minimal elements	No extraneous readouts; the γ readout appears only in Layer 2.
B1	Real terminology	"velocity", "energy", "force", "Newtonian prediction" — no invented labels.
B3	Symbols defined on first use	F, p, γ, m, v, c, E are defined in the Inquiry panel and on axes.
C1	Numbers referenced	Force value, v/c, E/mc², and γ are each tied to a prompt or annotation.
C2	Frame + units	Axes read "v / c (lab frame)", "E / mc² (lab frame)"; F shown in c/s.
D3	Shape + label	Solid coloured "relativistic" curve vs dashed grey "Newtonian prediction".
D6	Additive layering	Layers 0→1→2 add shading/annotations; never remove controls.
D7	No gating	Slider, Play/Pause, prediction all live from first load.
D8	Canvas labels	Axes, curve labels and asymptote live on the canvas; prose in panels.
E1	Faithful evolution	Traces build monotonically; v never exceeds c; E never plateaus.
E3	No decorative extras	Clean traces and an asymptote line — no sparkles/glow.
F2	Conceptual prediction	"What shape will the curve have?" — qualitative, not numeric.
F9–F14	Prompt wording	Open eliciting prompt; formative feedback ("where does the missing velocity go?").
H1	Structural misconception	The Newtonian dashed line visually diverges and exits the plot.
H2	Misconception as choice	Option (A) is Alex's claim; rebuttal appears only when chosen.
H3	Persona	"Alex says…" concept cartoon; the student argues against it.
J2	Display honesty	No hidden normalisation; axes explicitly v/c and E/mc².
J3	Correct model	Exact analytic v(t), E(t) — not naïve Euler of F = ma.
J5	Stable axes	v-axis fixed 0–1.2; energy axis fixed once per force (never per-frame).

Teach me

Speed

Guided Inquiry

1 · A thrust that never stops

Let’s begin with a push that never lets up. A body at rest is pushed by a steady force F, never switched off. Newton’s F = ma gives a straight line rising for ever. So what do you think the speed really does?

Now press Play and compare the relativistic curve (solid) with the Newtonian prediction (dashed). How did your guess hold up?

2 · Where the work goes

Here’s a puzzle for you. The force never changes and the body never stops moving, so work is done at every instant. Yet late in the push the speed barely changes. What do you think all that work is buying?

Now turn on ② Connect to energy, and you’ll see the γ readout appear beside the energy trace.

3 · Below one-fifth of c

Now let’s come back down to everyday speeds. Cars, rockets and planets obey Newton’s F = ma to superb accuracy. Take the same steady force but stop the record at 0.2 c. How far apart do you think the two speeds are there?

Try it yourself: press ⊙ View low velocity, then Play. The solid curve runs only to 0.2 c and lies almost on the dashed line.

4 · Doubling the force

One last experiment. At F/m = 0.5 c/s the speed reads 0.981 c ten seconds in. Double the force to F/m = 1.0 c/s and start again. What’s your guess? Ten seconds in, the speed now reads…

See for yourself: slide F to 1.0 and press Play, and watch the speed badge pass 0.995 c a third of the way across.

5 · Hyperbolic motion

Let’s put a name to everything you’ve just seen. The speed limit isn’t added by hand: it follows from F = dp/dt once momentum is p = γmv — a steady force adds momentum for ever, but the speed that momentum buys saturates at c while E = γmc² grows without bound. Here a₀ = F/m is what an accelerometer on board would read, the proper acceleration; a worldline at constant proper acceleration is a hyperbola in spacetime, hence hyperbolic motion.

Listen

Controls

▾

Applied force F

a₀ = F/m = 0.50 c/s

Plot layers

Show energy plot

Newton's 2nd law in Lorentz-invariant form: force = rate of change of relativistic momentum.

Lorentz factor — diverges as v → c, which is what stops the speed-up.

Velocity for constant F starting from rest — strictly bounded by c (solid curve).

Total energy = γmc² — grows without bound (lower plot).

Newtonian prediction (dashed grey) — an unbounded straight line.

Long-time limits: speed saturates at c, energy diverges. That is the whole insight.

The Physics

Relativistic constant acceleration · what this sim shows, the physics behind it, and every control

At a glance

This sim pushes a body from rest with a steady force that is never switched off, and plots its speed and energy against time beside Newton's F = ma prediction. It shows what relativity changes, and why that matters in particle accelerators.

What you'll learn

How Newton's second law is written to hold at any speed.
How speed and energy depend on time under a constant force.
Where Newton's F = ma works well, and where it stops.
What changing the force does to speed, momentum and energy.

A body of mass m at rest; a force F is applied and never removed.

What's on screen

Relativistic speed (solid cyan, upper panel): v/c, with a badge.
Newtonian prediction (dashed grey): v = Ft/m, with a triangle where it leaves the plot.
v = c (dashed rose): the line v/c = 1.
Energy (solid amber, lower panel): E in mc², with a badge. The dashed line is E = mc².

Units: c = 1 and m = 1. Speeds are in c, F is shown as a₀ = F/m in c per second, and t is lab time in seconds.

See itRun the body from rest at your force. Watch for: the solid trace and the dashed grey line leaving the origin together, then parting.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Applied force F	The steady force, shown as a₀ = F/m in c per second. Changing it restarts the run.	0.2–1.0 c/s opens at 0.5
① Highlight divergence (Plot layers)	Shades the gap between the solid trace and the dashed line, and labels it Δv.	on / off
② Connect to energy (Plot layers)	Adds a γ badge beside the energy trace, and switches the energy plot on.	on / off
⊙ View low velocity (Plot layers)	Zooms the time axis to 0 – 1.2/a₀ s. The solid trace and the energy stop at v = 0.2 c. It hides the energy plot, rewinds the run and pauses.	on / off
Show energy plot	Shows or hides the lower panel with E/mc².	on / off opens on
Badges (on the plot)	v (in c) and E (in mc²) at the current time, in the lab frame. With ② on, γ as well.	on the plot
▶ Play / ⏸ Pause	Runs time forward, or freezes the run so you can read the badges. The run ends at t = 30 s.	top bar
Speed	Plays the run faster or slower on screen. The physics stays the same.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning. F returns to 0.5 c/s and the run rewinds to t = 0.	top bar

Fixed: the body starts at rest, and the plot covers 30 s of lab time.

A steady force adds momentum at a steady rate

Relativity keeps Newton's second law in the form F = dp/dt, and changes the momentum to p = γmv.

Force is the rate of change of relativistic momentum.

A constant F acting from rest gives p = Ft, a straight line in time. In the sim's units, p/mc = a₀t/c.

The dashed grey line, v = Ft/m, is therefore also p/m: the momentum the force has delivered, divided by m.

Key idea. A constant force fixes how fast the momentum grows, dp/dt. It does not fix how fast the speed grows.

See itRun the body and follow the dashed grey line. Watch for: the dashed line rising straight and leaving the top of the plot at t = 1.2/a₀ (2.4 s at a₀ = 0.5 c/s), marked by a grey triangle.

Why the speed cannot keep up

Solve p = γmv for v, with p = Ft. Write x = p/mc = a₀t/c. Then v/c = x/√(1+x²):

The speed bought by a steady force, from rest.

This stays below 1 however large x gets. The factor γ = 1/√(1−v²/c²) grows without limit as v approaches c, so p = γmv can keep growing while v changes less and less.

v/c against p/mc, for the sim's law. At a₀ = 0.5 c/s, p/mc = 1, 5, 15 are t = 2, 10, 30 s.

This slider moves the sim's F slider too.

The lab acceleration is not constant: dv/dt = a₀/γ³, which falls toward zero as γ grows.

See itRun the body to the end of the plot. Watch for: the v badge creeping toward 1 c without reaching it, long after the dashed line has left the plot.

Newton's prediction at low speed

At a given time both laws have the same momentum, p = Ft. Newton's speed is p/m. The relativistic speed is p/(γm), so Newton's prediction is too high by the factor γ.

For v well below c, γ ≈ 1 + v²/2c². Then p ≈ mv, and F = dp/dt becomes F = ma.

relativistic speed	Newton's prediction at the same t	too high by
0.2 c	0.204 c	2%
0.5 c	0.577 c	15%
0.9 c	2.06 c	129%

The error is γ − 1. It grows smoothly from zero: no speed switches relativity on.

See itZoom in on the first stretch, up to 0.2 c. Watch for: the solid trace lying almost on the dashed line until it stops at v = 0.200 c.

Where the work goes

The force keeps doing work, because the body keeps moving. The work raises the energy E = γmc², and E − mc² equals the work done so far.

Energy from the same solution. For large t, E ≈ Fct.

E and p are linked through the mass, and the mass does not change:

Signature (−, +, +, +).

The mass m is the same number in every frame. It does not grow with speed.
E and p each depend on the frame. The combination E² − p²c² does not, and it equals m²c⁴.
What grows is γ, and with it both p = γmv and E = γmc².

See itSwitch on ② Connect to energy and run the body. Watch for: the E and γ badges climbing together while the v badge barely changes.

What the force changes

The slider sets a₀ = F/m, and the run depends on the force only through a₀t. Doubling F doubles p at every time. A body of twice the mass under the same F follows the run of half the a₀.

at t = 10 s	p/mc	v	E
a₀ = 0.5 c/s	5	0.981 c	5.10 mc²
a₀ = 1.0 c/s	10	0.995 c	10.05 mc²

Twice the force gives twice the momentum and almost twice the energy. The speed rises by only 0.014 c.

See itRun to t = 10 s at a₀ = 0.5 c/s, then again at 1.0 c/s. Watch for: each run pausing at 10 s, and the card showing both speeds side by side.

Where it matters

Four real beams, from γ = E/mc² (or 1 + K/mc²). γ spans five decades, and v/c stays below 1.

Accelerators. A proton in the LHC at 6.8 TeV has γ ≈ 7247 and v/c = 1 − 9.5 × 10⁻⁹, about 3 m/s short of c. Raising the beam energy hardly changes the speed, which is why a collider is rated by energy.
Medical linacs. A 6 MeV electron already has γ = 12.7 and v = 0.9969 c. Its energy keeps rising along the machine while its speed barely moves.
Old television tubes. A 20 keV electron in a CRT has v = 0.27 c and γ = 1.04, so Newton's p = mv is already about 4% low.

Go deeper: the view from on board

For a force along the motion, the force is the same in the lab and in the body's instantaneous rest frame. So a₀ = F/m is what an accelerometer on board reads, and it stays constant. The sim plots only lab-frame quantities.

The worldline is a hyperbola. τ is the proper time on the onboard clock.

The default run (a₀ = 0.5 c/s) as a worldline, from the formulas. At lab time t = 10 s the onboard clock reads τ = 4.62 s.

Summary

In one line. A constant force adds momentum at a constant rate, but the speed that momentum buys is capped by c, so the work goes into the energy E = γmc².

Key points

F = dp/dt with p = γmv, so from rest p = Ft.
v/c = x/√(1+x²), with x = a₀t/c. It stays below 1, and matches Newton's Ft/m only at low speed.
E = γmc² keeps rising, about Fct for large t. The mass stays the same.
The force enters through a₀ = F/m. Doubling it doubles p, but barely changes v near c.

Common mix-ups

"A constant force means a constant acceleration, so the speed rises for ever." A constant F fixes dp/dt. The lab acceleration is a₀/γ³, which falls toward zero.
"The body gets heavier, and that is what stops it." The mass is the same in every frame. What grows is γ.
"Relativity only matters near c." γ − 1 is already about 2% at 0.2 c, and grows smoothly from zero.

Things to try on your own

Set F to 0.2, 0.6 and 1.0 in turn, pause at t = 10 s and note the v badge each time.
At a₀ = 0.2 c/s, press ⊙ and find when the solid trace stops. Check it against t = 0.204/a₀ ≈ 1.02 s.
Switch on ② and pause at t = 20 s with a₀ = 0.5 c/s. Compare the E badge with γ = √(1 + (a₀t/c)²).